This could perhaps be accomplished with a basic excel sheet or just keeping notes about you characters advancement in LI stats, but, well...

I use Alt Inventory to keep track of what I have in Bags and Vaults, but is it possible to keep track of your Characters Gear, Stats and LI items, down to individual Relics slotted to LI's somehow?

Since I do a lot of crafting I also do Crafted Relics that I exchange for Insignias or Devices at the Guild Master and more than once I end up with either the wrong item or duplicates which I dump to AH. Something in game helping to keep track of this would be very useful since the more Characters you have, the more difficult it becomes to keep track of these things.

You could only do that with a pluginn where you input everything yourself. Plugins have no access to data about the characters LIs other then what is in the characters inventory, and that only gives you item type and name, no lvls, relics and such.

I am not using this myself, but I saw people using Google Sheets on a second monitor. If you decide to track you have the create the habit/discipline of taking notes.

Personally I would amass resources beforehand, build the perfect LI for a class, and then continue amassing ressources until the next one was about to be built/leveled etc. That way I only had to plan out an LI in advance, and remember what LI I want to build next. Not the exact stats of all LIs.

For stuff like anfalas crystals, I would just store them in account storage, and use them ASAP on LIs on the most played characters. So my champion main until she had perfect LIs, then the runekeeper, then my bard, etc.